Can I get a new random number per ray in the node editor?
There's enough randomness in the Position from the Geometry node. It will almost never be exactly the same for two samples, but you have to apply a hash function to convert it to a random number in the [0,1) range.

The simplest solution would be to use a Voronoi or Noise Texture node with a very large Scale.

Another solution is to make your own hash function using math nodes, replicating for example this GLSL code.